Problem: Solve for $x$ : $ 5|x + 7| - 5 = -5|x + 7| + 7 $
Explanation: Add $ {5|x + 7|} $ to both sides: $ \begin{eqnarray} 5|x + 7| - 5 &=& -5|x + 7| + 7 \\ \\ { + 5|x + 7|} && { + 5|x + 7|} \\ \\ 10|x + 7| - 5 &=& 7 \end{eqnarray} $ Add ${5}$ to both sides: $ \begin{eqnarray} 10|x + 7| - 5 &=& 7 \\ \\ { + 5} &=& { + 5} \\ \\ 10|x + 7| &=& 12 \end{eqnarray} $ Divide both sides by ${10}$ $ \dfrac{10|x + 7|} {{10}} = \dfrac{12} {{10}} $ Simplify: $ |x + 7| = \dfrac{6}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 7 = -\dfrac{6}{5} $ or $ x + 7 = \dfrac{6}{5} $ Solve for the solution where $x + 7$ is negative: $ x + 7 = -\dfrac{6}{5} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& -\dfrac{6}{5} \\ \\ {- 7} && {- 7} \\ \\ x &=& -\dfrac{6}{5} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{6}{5} {- \dfrac{35}{5}} $ $ x = -\dfrac{41}{5} $ Then calculate the solution where $x + 7$ is positive: $ x + 7 = \dfrac{6}{5} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& \dfrac{6}{5} \\ \\ {- 7} && {- 7} \\ \\ x &=& \dfrac{6}{5} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{6}{5} {- \dfrac{35}{5}} $ $ x = -\dfrac{29}{5} $ Thus, the correct answer is $x = -\dfrac{41}{5} $ or $x = -\dfrac{29}{5} $.